Young giant exoplanets are a unique laboratory for understanding cool, low-gravity atmospheres. A quintessential example is the massive extrasolar planet $beta$ Pic b, which is 9 AU from and embedded in the debris disk of the young nearby A6V star $beta$ Pictoris. We observed the system with first light of the Magellan Adaptive Optics (MagAO) system. In Paper I we presented the first CCD detection of this planet with MagAO+VisAO. Here we present four MagAO+Clio images of $beta$ Pic b at 3.1 $mu$m, 3.3 $mu$m, $L^prime$, and $M^prime$, including the first observation in the fundamental CH$_4$ band. To remove systematic errors from the spectral energy distribution (SED), we re-calibrate the literature photometry and combine it with our own data, for a total of 22 independent measurements at 16 passbands from 0.99--4.8 $mu$m. Atmosphere models demonstrate the planet is cloudy but are degenerate in effective temperature and radius. The measured SED now covers $>$80% of the planets energy, so we approach the bolometric luminosity empirically. We calculate the luminosity by extending the measured SED with a blackbody and integrating to find log($L_{bol}$/$L_{Sun}$) $= -3.78pm0.03$. From our bolometric luminosity and an age of 23$pm$3 Myr, hot-start evolutionary tracks give a mass of 12.7$pm$0.3 $M_{Jup}$, radius of 1.45$pm$0.02 $R_{Jup}$, and $T_{eff}$ of 1708$pm$23 K (model-dependent errors not included). Our empirically-determined luminosity is in agreement with values from atmospheric models (typically $-3.8$ dex), but brighter than values from the field-dwarf bolometric correction (typically $-3.9$ dex), illustrating the limitations in comparing young exoplanets to old brown dwarfs.
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